Elastic waves in bodies with initial stresses

Conclusions An analysis of research available on the problem under consideration shows that at this time: a) Wave propagation in unbounded, bounded, and composite laminar bodies with homogeneous initial states has been studied sufficiently extensively; b) methods to determine the third-order elastic constants have been developed on the basis of existing theories; c) an ultrasonic nondestructive method to determine the stress in solids has been developed, where the stresses averaged over the bulk of the body are determined.In light of the above, in the authors' opinion, the following should be considered the most urgent questions for further investigations on the problem: a) the investigation of wave propagation regularities in bodies with inhomogeneous initial states (it is hence important to execute quantitative and qualitative analyses of the phenomenon); b) development of an ultrasonic nondestructive method of determining the stress in the near surface layers of a solid, which will afford a possibility of determining not only the membrane stresses but also the bending stresses; c) an investigation of the wave propagation regularities in fibrous composites with initial stresses.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 15, No. 4, pp. 3–23, April, 1979.     

Elastic waves in a turbulent fluid

It is shown that a turbulent flow can support elastic waves for which the fluctuating stress is essentially proportional to the strain field. These fluctuations are described by a hyperbolic system and hence they do not decay away from a solid boundary. Their behaviour is also consistent with Taylor's frozen field hypothesis that turbulent fluctuations are advected with the local mean flow speed.     

Waves in a prestressed elastic layer in compression interacting with an ideal fluid

S. P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 3–10, February, 1994.     

The generalized Boussinesq equations and obliquely interacting solitary waves in a stratified fluid

In this paper, on the basis of Boussinesq’s shallow water theory, we establish the basic equations governing the motion of a stratified fluid, a kind of the generalized Boussinesq equations. And then by way of them, we study the weak interaction of two pairs of obliquely colliding solitary waves, give the second-order approximate solutions for wave profiles and maximum amplitudes, as well as conclude that when the included angle between the directions of propagation of impinging solitary waves is less than 120°, the effect of oblique interaction is stronger than that of the head-on one, but when the angle concerned is greater than 120°, the former is slightly weaker than the latter.     

Influence of fluid viscosity on waves in an initially deformed,compressible, elastic layer interacting with a fluid medium

The propagation of harmonic waves along the interface of an initially stressed, compressible layer and a viscous, compressible fluid half-space is investigated. A dispersion relation that does not depend on the form of the elastic potential is derived on the basis of the three-dimensional linearized elasticity equations for elastic bodies with uniform initial deformations and on the linearized Navier-Stokes equations for a viscous Newtonian fluid at rest. The phase velocities and attenuation coefficients of the elastic modes are determined numerically as functions of the thickness of the elastic layer using a Murnaghan-type three-invariant elastic potential.S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 9, pp. 3–9, September, 1994.     

Elastic waves in a Timoshenko beam with boundary damping

In this paper we consider a Timoshenko beam with a damping moment applied to one of the endpoints. The elastic waves that develop from two sets of localised initial disturbances in the beam, are simulated and it is shown that properties of the spectrum adequately explain the features of the waves. We also show, using energy calculations, that the boundary moment is significantly more effective in reducing vibrations in the beam in one of the cases under consideration. The so-called second spectrum of the Timoshenko beam plays a prominent part in explaining this phenomenon.     

Elastic waves in a fiber-reinforced composite

The propagation of time-harmonic elastic waves in a fiber-reinforced composite is studied. The circular fibers are assumed to be parallel to each other and randomly distributed with a statistically uniform distribution. The direction of propagation and the associated particle motion are considered to be normal to the fibers. It is shown that the average waves in the composite separate into compressional and shear types. General formulae for the complex wave number giving the phase velocity and the damping are obtained. It is shown that these formulae lead to the Hashin-Rosen expressions for the transverse bulk modulus and the lower bound for the transverse rigidity, if the correlation in the positions of the fibers can be ignored. The correlation terms, for exponential correlation, are shown to have a significant effect on the damping property of the composite, especially at high frequencies and concentrations.     

Elastic bodies reinforced with inextensible surfaces

The constraint of in-plane rigidity is examined within the general framework of the theory of internally constrained materials. It is shown that, for in-plane rigid materials, local strain and active stress are both defined by vectorial quantities. Representation formulae for the elastic response mapping are established in the cases of transverse isotropy and maximal symmetry, compatible with the constraint manifold. The equilibrium problem for an elastic body reinforced with parallel inextensible planes is also considered. In particular, universal solutions for bodies with maximal material symmetry are determined within the class of deformations which leave rigid every reinforcing plane.     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.     

Elastic waves against a corrugated boundary

A train of plane waves travels in an elastic semi-infinite medium bounded by a corrugated line having a sinusoidal shape. When the primary waves impinge against the surface, a new train of reflected waves is generated, and the question arises of determining the effect of roughnesses of the boundary on the shape and amplitude of the reflected waves.The case of perpendicular incidence may be treated without difficulty by extending a solution found by Rayleigh [2, Art. 272] for reflection of sound waves from a corrugated surface.

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Sommario Un treno d'onde piane viaggia in un mezzo elastico semi-indefinito limitato da una linea corrugata di forma sinusoidale. Quando l'onda primaria urta contro la frontiera, si genera un nuovo treno d'onde, e si pone la questione di determinare l'effetto della rugosità del contorno sulla forma e l'ampiezza delle onde rillesse.Il caso di incidenza perpendicolare si può trattare senza difficoltà estendendo una soluzione trovata da Rayleigh [2, Art. 272] sulla riflessione delle onde sonore da parte di una superficie corrugata.

     

Acceleration waves in a fluid mixture with compressible constituents

An analysis is presented for the problem of propagating acceleration waves in a binary, immiscible fluid mixture with compressible constituents. A purely mechanical theory is used to describe the behavior of the mixture, and the local volume fractions (concentrations) of the constituents appear among the independent variables in constitutive equations. These volume fractions are constrained by the condition that they sum to unity. The theory of singular surfaces is used to analyse the propagation of waves. A quartic equation is obtained whose solutions are the permissible wave speeds. Conditions for the growth or decay of the amplitude of the wave are also given.     

Nonlinear waves in a viscous compressible fluid with relaxation

The propagation and stability of nonlinear waves in a viscous compressible fluid with relaxation that satisfies a Theological equation of state of Oldroyd type are investigated. An equation that describes the structure of the wave perturbations and its evolution is derived subject to the condition of balance of the nonlinear dissipative and relaxation effects, and its solutions of the solitary wave type are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 31–35, May–June, 1993.     

Localized strain waves in a nonlinearly elastic conducting medium interacting with a magnetic field

The influence of magnetic field on the generation of a localized wave in a nonlinearly elastic conducting medium is considered. The evolution equation for describing the wave beam propagation in the medium is derived. It is shown that the wave beam parameters depend on the value of the external magnetic field and on the field orientation in space.     

Dynamic loading of cylindrical and spherical bodies interacting with a shock wave

Empirical formulas for diffraction pressure on the surface of a ball and a cylinder interacting with a weak shock wave are derived based on experimental data.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 117–123, December 2004.